Taking two new-born children belonging to persons of the common sort he gave them to a shepherd to bring up at the place where his flocks were, with a manner of bringing up such as I shall say, charging him namely that no man should utter any word in their presence, and that they should be placed by themselves in a room where none might come . . .They would be fed and otherwise cared for but no one would talk to them. Psammetichos wanted to hear what the first words would be from the children. After a couple of years, the children started saying 'bekos'. When this was reported to the king, some effort was made to find out if that was a word in any language. It was discovered that 'bekos' meant 'bread' to the Phrygians and they were declared the earliest people.
Obviously there are some ethical issues with taking children away and doing experiments with them, but set that aside. The king tried to create a sterile experiment so that he could find a true result. And when that result came out in favor a different people, he still allowed that result to be known.
Herodotus also mentions Egyptian astronomy. They were the first people to calculate the year, having twelve months of 30 days with five extra days. Herodotus says this was better than the Hellenic manner where extra months were needed to make it all come out right. (Bonus question: why are months standardized at 30 days?)
Herodotus also questioned the source of the Nile. He knew that rivers usually (always?) start with melted snow. He disregarded this option for the Nile since it flowed from the desert where there was no snow. Of course we now know that it simply flows through the Sahara and starts quite a bit south of there. The source wasn't found for more than two millennia later so I don't mean that as any kind of criticism. It's hard not to be impressed with how he grappled with the problem and tried to solve it.
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